Discrete spectrum of a model Schrödinger operator on the half-plane with Neumann conditions
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We study the eigenpairs of a model Schrödinger operator with a quadratic potential and Neumann boundary conditions on a half-plane. The potential is degenerate in the sense that it reaches its minimum all along a line that makes the angle θ with the boundary of the half-plane. We show that the first eigenfunctions satisfy localization properties related to the distance to the minimum line of the potential. We investigate the densification of the eigenvalues below the essential spectrum in the limit θ → 0, and we prove a full asymptotic expansion for these eigenvalues and their associated eigenvectors. We conclude the paper by numerical experiments obtained by a finite element method. The numerical results confirm and enlighten the theoretical approach.
Mathematics Subject Classification (2000)Primary 99Z99 Secondary 00A00 35P15 35P20 41A60 65N25 65N30
KeywordsAgmon estimates Born-Oppenheimer approximation Schrödinger operator Semi classical limit
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